and for any \(k,l\in \lbrace 1,\dots , N\rbrace \)  with \(k\ne l\) ,
we have \({\cal E}^{(c)}(\varphi _n,\varphi _l)=0\) 
by the strong local property ([1]}).
Hence by following the proof of [2]}
and using (REF ),
there exists \(c_1>0\)  such that
\(\begin{split}&{\cal E}(\varphi _n,\varphi _n)+2\sum _{l=n+1}^{N}|{\cal E}(\varphi _n,\varphi _l)|\\&\le \frac{1}{(s_n-s_{n-1})^2}M_0(B(s_n))\\&+\frac{1}{s_n-s_{n-1}}\left(\frac{1}{s_n-s_{n-1}}+\frac{2}{s_{n+1}-s_n}\right)M_1(s_{n+1})+10\sum _{l=n+1}^{N-1}J(B(s_n)\times B(s_l)^c)\\&\le c_1\left\lbrace \frac{1}{s_n^2}(M_0(s_n)+M_1(s_{n+1}))+M_2(s_n,c_0s_n)\right\rbrace .\end{split}\) 
